Abstract
Joint maximum likelihood estimation (JMLE) is developed for diagnostic classification models (DCMs). JMLE has been barely used in Psychometrics because JMLE parameter estimators typically lack statistical consistency. The JMLE procedure presented here resolves the consistency issue by incorporating an external, statistically consistent estimator of examinees’ proficiency class membership into the joint likelihood function, which subsequently allows for the construction of item parameter estimators that also have the consistency property. Consistency of the JMLE parameter estimators is established within the framework of general DCMs: The JMLE parameter estimators are derived for the Loglinear Cognitive Diagnosis Model (LCDM). Two consistency theorems are proven for the LCDM. Using the framework of general DCMs makes the results and proofs also applicable to DCMs that can be expressed as submodels of the LCDM. Simulation studies are reported for evaluating the performance of JMLE when used with tests of varying length and different numbers of attributes. As a practical application, JMLE is also used with “real world” educational data collected with a language proficiency test.
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References
Baker, F. B., & Kim, S.-H. (2004). Item response theory: Parameter estimation techniques (2nd ed.). New York: Marcel Dekker.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F. M. Load & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 397–479). Reading, MA: Addison-Wesley.
Buck, G., & Tatsuoka, K. K. (1998). Application of the rule-space procedure to language testing: Examining attributes of a free response listening test. Language Testing, 15, 119–157.
Chiu, C.-Y., & Douglas, J. A. (2013). A nonparametric approach to cognitive diagnosis by proximity to ideal response profiles. Journal of Classification, 30, 225–250.
Chiu, C.-Y., Douglas, J. A., & Li, X. (2009). Cluster analysis for cognitive diagnosis: Theory and applications. Psychometrika, 74, 633–665.
Chiu, C.-Y., & Köhn, H.-F. (2015). Consistency of cluster analysis for cognitive diagnosis: The DINO model and the DINA model revisited. Applied Psychological Measurement, 39, 465–479.
de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179–199.
de la Torre, J., & Chiu, C.-Y. (2010, April). A general empirical method of Q-matrix validation. Paper presented at the annual meeting of the National Council on Measurement in Education, Denver, CO.
DiBello, L. V., Roussos, L. A., & Stout, W. F. (2007). Review of cognitively diagnostic assessment and a summary of psychometric models. In C. R. Rao & S. Sinharay (Eds.), Handbook of Statistics. Psychometrics (Vol. 26, pp. 979–1030). Amsterdam: Elsevier.
ECPE 2013 report. (2013). Retrieved March 10, 2013, from http://www.cambridgemichigan.org/resources/ecpe/reports
Embretson, S. E., & Reise, S. P. (2000). Item response theory for psychologists. Mahwah, NJ: Erlbaum.
Feng, Y., Habing, B. T., & Huebner, A. (2014). Parameter estimation of the Reduced RUM using the EM algorithm. Applied Psychological Measurement, 38, 137–150.
Haberman, S. J. (2004, May). (2005, September). Joint and conditional maximum likelihood estimation for the Rasch model for binary responses. Research report no. RR-04-20. Princeton, NJ: Educational Testing Service.
Haberman, S. J., & von Davier, M. (2007). Some notes on models for cognitively based skill diagnosis. In C. R. Rao & S. Sinharay (Eds.), Handbook of statistics. Psychometrics (Vol. 26, pp. 1031–1038). Amsterdam: Elsevier.
Hartz, S. M. (2002). A Bayesian framework for the Unified Model for assessing cognitive abilities: Blending theory with practicality. Doctoral dissertation. Available from ProQuest Dissertations and Theses database. UMI No. 3044108.
Hartz, S. M., & Roussos, L. A. (October 2008). The Fusion Model for skill diagnosis: Blending theory with practicality. Research report no. RR-08-71. Princeton, NJ: Educational Testing Service.
Henson, R. A., & Templin, J. (2007, April). Large-scale language assessment using cognitive diagnosis models. Paper presented at the annual meeting of the National Council on Measurement in Education, Chicago, IL.
Henson, R. A., Templin, J. L., & Willse, J. T. (2009). Defining a family of cognitive diagnosis models using log-linear models with latent variables. Psychometrika, 74, 191–210.
Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301), 13–30.
Junker, B. W. (1991). Essential independence and likelihood-based ability estimation for polytomous items. Psychometrika, 56, 255–278.
Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25, 258–272.
Leighton, J., & Gierl, M. (2007). Cognitive diagnostic assessment for education: Theory and applications. Cambridge, UK: Cambridge University Press.
Leighton, J. P., Gierl, M. J., & Hunka, S. (2004). The attribute hierarchy model: An approach for integrating cognitive theory with assessment practice. Journal of Educational Measurement, 41, 205–236.
Liu, Y., Douglas, J. A., & Henson, R. A. (2009). Testing person fit in cognitive diagnosis. Applied Psychological Measurement, 33, 579–598.
Lunn, D., Spiegelhalter, D., Thomas, A., & Best, N. (2009). The BUGS project: Evolution, critique, and future directions. Statistics in Medicine, 28, 3049–3067.
Macready, G. B., & Dayton, C. M. (1977). The use of probabilistic models in the assessment of mastery. Journal of Educational Statistics, 33, 379–416.
Muthén, L. K., & Muthén, B. O. (1998–2012). Mplus user’s guide (7th edn.) Los Angeles: Muthén & Muthén.
Neyman, J., & Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1–32.
Robitzsch, A., Kiefer, T., George, A. C., & Uenlue, A. (2015). CDM: Cognitive diagnosis modeling. R package version 3.1-14. Retrieved from the Comprehensive R Archive Network [CRAN] website
Rupp, A. A., Templin, J. L., & Henson, R. A. (2010). Diagnostic measurement: Theory, methods, and applications. New York: Guilford.
Tatsuoka, K. (1985). A probabilistic model for diagnosing misconception in the pattern classification approach. Journal of Educational Statistics, 12, 55–73.
Templin, J. L., & Henson, R. A. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11, 287–305.
Templin, J., & Bradshaw, L. (2014). Hierarchical diagnostic classification models: A family of models for estimating and testing attribute hierarchies. Psychometrika, 79, 317–339.
Templin, J., & Hoffman, L. (2013). Obtaining diagnostic classification model estimates using Mplus. Educational Measurement: Issues and Practice, 32, 37–50.
Vermunt, J. K., & Magidson, J. (2000). Latent GOLD’s users’s guide. Boston: Statistical Innovations Inc.
von Davier, M. (2005, September). A general diagnostic model applied to language testing data. Research report no. RR-05-16. Princeton, NJ: Educational Testing Service.
von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287–301.
Wang, S., & Douglas, J. (2015). Consistency of nonparametric classification in cognitive diagnosis. Psychometrika, 80, 85–100.
Zheng, Y., & Chiu, C.-Y. (2014). NPCD: Nonparametric methods for cognitive diagnosis. R package version 1.0-5. http://CRAN.R-project.org/package=NPCD
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Chiu, CY., Köhn, HF., Zheng, Y. et al. Joint Maximum Likelihood Estimation for Diagnostic Classification Models. Psychometrika 81, 1069–1092 (2016). https://doi.org/10.1007/s11336-016-9534-9
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DOI: https://doi.org/10.1007/s11336-016-9534-9